Group #9862 — subgroups

Subgroups — #9862

Order 256.

257 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 257 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3147	C₂	1	1	yes	maximal, characteristic
2	#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	#1492	C₂	1	4	yes	maximal, complemented
4	C₂×#322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#135	—	2	2	no	—
6	#532	C₂²	1	2	yes	—
7	#495	—	2	2	no	—
8	#494	C₄	1	1	yes	characteristic
9	#493	C₄	1	1	yes	characteristic
10	#492	C₄	1	2	yes	—
11	C₂×#108	—	2	2	no	—
12	#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	D₄⋊C₈	—	2	8	no	—
16	#322	—	2	2	no	—
17	C₂×#41	—	2	2	no	—
18	#135	—	4	4	no	—
19	#135	—	4	4	no	—
20	C₂×Q₁₆	—	2	2	no	—
21	C₂×SD₁₆	—	2	4	no	—
22	C₂×D₈	—	2	2	no	—
23	C₂×M₁₆	—	2	2	no	—
24	C₂²×C₈	—	2	2	no	—
25	C₂²×C₈	D₄	1	2	yes	—
26	C₄⋊Q₈	C₂×C₄	1	2	yes	—
27	#127	C₂×C₄	1	2	yes	—
28	#124	—	2	2	no	—
29	#121	—	2	4	no	—
30	C₄×D₄	—	2	4	no	—
31	C₂×C₄²	C₂³	1	1	yes	characteristic
32	#108	—	4	4	no	—
33	C₄⋊C₈	—	2	4	no	—
34	C₄×C₈	—	2	2	no	—
35	C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈	D₄	1	4	yes	—
38	C₄×C₈	D₄	1	2	yes	—
39	#41	—	4	4	no	—
40	#41	—	4	4	no	—
41	C₂×Q₈	—	2	2	no	—
42	C₂×D₄	—	2	2	no	—
43	C₂×D₄	—	2	4	no	—
44	C₂×D₄	—	4	2	no	—
45	C₂²×C₄	—	4	2	no	—
46	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄	#41	1	2	yes	—
48	Q₁₆	—	4	4	no	—
49	SD₁₆	—	4	8	no	—
50	D₈	—	4	4	no	—
51	M₁₆	—	4	4	no	—
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	4	4	no	—
54	C₂×C₈	—	4	2	no	—
55	C₂×C₈	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈	D₈	1	4	yes	—
58	C₂×C₈	SD₁₆	1	4	yes	—
59	C₄⋊C₄	—	2	4	no	—
60	C₂²⋊C₄	—	4	4	no	—
61	C₄²	C₂²×C₄	1	1	yes	characteristic
62	C₄²	C₂×D₄	1	1	yes	characteristic
63	C₄²	C₂×D₄	1	2	yes	—
64	C₂³	—	4	2	no	—
65	C₂³	#124	1	1	yes	characteristic
66	Q₈	—	4	4	no	—
67	D₄	—	4	4	no	—
68	D₄	—	4	8	no	—
69	D₄	—	8	4	no	—
70	C₂×C₄	—	2	4	no	—
71	C₂×C₄	—	4	2	no	—
72	C₂×C₄	—	8	2	no	—
73	C₂×C₄	—	8	2	no	—
74	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄	C₄×D₄	1	2	yes	—
76	C₂×C₄	#121	1	2	yes	—
77	C₂×C₄	#127	1	1	yes	characteristic
78	C₂×C₄	C₂×D₈	1	2	yes	—
79	C₂×C₄	C₂×SD₁₆	1	2	yes	—
80	C₈	—	4	4	no	—
81	C₈	—	8	2	no	—
82	C₈	#102	1	4	yes	—
83	C₈	#102	1	4	yes	—
84	C₂²	—	2	2	no	—
85	C₂²	—	8	2	no	—
86	C₂²	—	8	4	no	—
87	C₂²	#390	1	1	yes	characteristic
88	C₂²	#486	1	2	yes	—
89	C₄	—	2	2	no	—
90	C₄	—	8	2	no	—
91	C₄	C₂×#102	1	2	yes	—
92	C₄	#444	1	4	yes	—
93	C₄	#492	1	1	yes	characteristic
94	C₄	#493	1	1	yes	characteristic
95	C₂	—	2	2	no	—
96	C₂	—	16	2	no	—
97	C₂	#1957	1	1	yes	characteristic
98	C₂	#1975	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3147

C₂

yes

maximal, characteristic

#1848

C₂

yes

maximal, characteristic, complemented

#1492

C₂

yes

maximal, complemented

C₂×#322

C₂

yes

maximal, characteristic, complemented

C₂×#135

—

#532

C₂²

yes

—

#495

—

#494

C₄

yes

characteristic

#493

C₄

yes

characteristic

#492

C₄

yes

—

C₂×#108

—

#423

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

D₄⋊C₈

—

#322

—

C₂×#41

—

#135

—

#135

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

C₂×C₄

yes

—

#127

C₂×C₄

yes

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂³

yes

characteristic

#108

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

—

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

D₈

yes

—

C₂×C₈

SD₁₆

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

—

C₂³

—

C₂³

#124

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

#121

yes

—

C₂×C₄

#127

yes

characteristic

C₂×C₄

C₂×D₈

yes

—

C₂×C₄

C₂×SD₁₆

yes

—

C₈

—

C₈

—

C₈

#102

yes

—

C₈

#102

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#390

yes

characteristic

C₂²

#486

yes

—

C₄

—

C₄

—

C₄

C₂×#102

yes

—

C₄

#444

yes

—

C₄

yes

C₄

yes

C₂

—

C₂

—

C₂

#1957

yes

characteristic

C₂

#1975

yes

—

	subgroup	G/H
center	C₂×C₄	#127
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#390
prime radical	#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#127

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

C₂²

C₂

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
2. #1848	C₂	semidirect
3. #1492	C₂	semidirect
4. C₂×#322	C₂	semidirect

H ≅ G/N

type

2. #1848

C₂

semidirect

3. #1492

C₂

semidirect

4. C₂×#322

C₂

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#3147	2
#1848	2
#1492	2
C₂×#322	2

subgroup

index

Subgroups — #9862

Conjugacy classes of subgroups

Complements

Maximal subgroups